Characterizations of bipartite and Eulerian partial duals of ribbon graphs

Huggett and Moffatt characterized all bipartite partial duals of a plane graph in terms of all-crossing directions of its medial graph. Then Metsidik and Jin characterized all Eulerian partial duals of a plane graph in terms of semi-crossing directions of its medial graph. Plane graphs are ribbon graphs with genus 0. In this paper, by introducing the notion of modified medial graphs and using their all-crossing directions, we first extend Huggett and Moffatt’s result from plane graphs to ribbon graphs. Then we characterize all Eulerian partial duals of any ribbon graph in terms of crossing-total directions of its medial graph, which are simpler than semi-crossing directions.     

Eulerian partial duals of plane graphs

In the paper Combinatorica 33(2) (2013) 231–252, Huggett and Moffatt characterized all bipartite partial duals of a plane graph in terms of oriented circuits in its medial graph. An open problem posed in their paper is the characterization of Eulerian partial duals of plane graphs. In this article, we solve this problem by considering half‐edge orientations of medial graphs.     

Rectangular-radial drawings of cubic plane graphs

Rectangular drawings and rectangular duals can be naturally extended to other surfaces. In this paper, we extend rectangular drawings and rectangular duals to drawings on a cylinder. The extended drawings are called rectangular-radial drawings and rectangular-radial duals. Rectangular-radial drawings correspond to periodic rectangular tilings of a 1-dimensional strip. We establish a necessary and sufficient condition for plane graphs with maximum degree 3 to have rectangular-radial drawings and a necessary and sufficient condition for triangulated plane graphs to have rectangular-radial duals. Furthermore, we present three linear time algorithms under three different conditions for finding a rectangular-radial drawing for a given cubic plane graph, if one exists.     

Forcing faces in plane bipartite graphs

Let Ω denote the class of connected plane bipartite graphs with no pendant edges. A finite face s of a graph G∈Ω is said to be a forcing face of G if the subgraph of G obtained by deleting all vertices of s together with their incident edges has exactly one perfect matching. This is a natural generalization of the concept of forcing hexagons in a hexagonal system introduced in Che and Chen [Forcing hexagons in hexagonal systems, MATCH Commun. Math. Comput. Chem. 56 (3) (2006) 649-668]. We prove that any connected plane bipartite graph with a forcing face is elementary. We also show that for any integers n and k with n?4 and n?k?0, there exists a plane elementary bipartite graph such that exactly k of the n finite faces of G are forcing. We then give a shorter proof for a recent result that a connected cubic plane bipartite graph G has at least two disjoint M-resonant faces for any perfect matching M of G, which is a main theorem in the paper [S. Bau, M.A. Henning, Matching transformation graphs of cubic bipartite plane graphs, Discrete Math. 262 (2003) 27-36]. As a corollary, any connected cubic plane bipartite graph has no forcing faces. Using the tool of Z-transformation graphs developed by Zhang et al. [Z-transformation graphs of perfect matchings of hexagonal systems, Discrete Math. 72 (1988) 405-415; Plane elementary bipartite graphs, Discrete Appl. Math. 105 (2000) 291-311], we characterize the plane elementary bipartite graphs whose finite faces are all forcing. We also obtain a necessary and sufficient condition for a finite face in a plane elementary bipartite graph to be forcing, which enables us to investigate the relationship between the existence of a forcing edge and the existence of a forcing face in a plane elementary bipartite graph, and find out that the former implies the latter but not vice versa. Moreover, we characterize the plane bipartite graphs that can be turned to have all finite faces forcing by subdivisions.     

Construction of some countable 1-arc-transitive bipartite graphs

We generalize earlier work which gave a method of construction for bipartite graphs which are obtained as the set of maximal or minimal elements of a certain cycle-free partial order. The method is extended here to produce a 1-arc-transitive bipartite graph in a ‘free’ way, starting with any partial order with greatest and least element and with instructions on its points about how they will ramify in the extension. A key feature of our work is the interplay between properties of the initial partial order, the extended partial order, and the bipartite graph which results. We also extend the earlier work by giving a complete characterization of all 2-CS-transitive cycle-free partial orders. In addition, we discuss the completeness of the constructed partial orders, in the sense of Dedekind and MacNeille, and remark that the bipartite graph constructed can only be 2-arc-transitive in the cycle-free case.     

Projective-planar graphs with even duals

Let G be a connected graph which is projective-planar but is not planar. It will be shown that G can be embedded in the projective plane so that it has only even faces if and only if either G is bipartite or its canonical bipartite covering is planar and that such an embedding is unique if G is 3-connected.     

On bipartite zero-divisor graphs

A (finite or infinite) complete bipartite graph together with some end vertices all adjacent to a common vertex is called a complete bipartite graph with a horn. For any bipartite graph G, we show that G is the graph of a commutative semigroup with 0 if and only if it is one of the following graphs: star graph, two-star graph, complete bipartite graph, complete bipartite graph with a horn. We also prove that a zero-divisor graph is bipartite if and only if it contains no triangles. In addition, we give all corresponding zero-divisor semigroups of a class of complete bipartite graphs with a horn and determine which complete r-partite graphs with a horn have a corresponding semigroup for r≥3.     

2-resonance of plane bipartite graphs and its applications to boron-nitrogen fullerenes

A set H of disjoint faces of a plane bipartite graph G is a resonant pattern if G has a perfect matching M such that the boundary of each face in H is an M-alternating cycle. An elementary result was obtained [Discrete Appl. Math. 105 (2000) 291-311]: a plane bipartite graph is 1-extendable if and only if every face forms a resonant pattern. In this paper we show that for a 2-extendable plane bipartite graph, any pair of disjoint faces form a resonant pattern, and the converse does not necessarily hold. As an application, we show that all boron-nitrogen (B-N) fullerene graphs are 2-resonant, and construct all the 3-resonant B-N fullerene graphs, which are all k-resonant for any positive integer k. Here a B-N fullerene graph is a plane cubic graph with only square and hexagonal faces, and a B-N fullerene graph is k-resonant if any disjoint faces form a resonant pattern. Finally, the cell polynomials of 3-resonant B-N fullerene graphs are computed.     

Excluded checkerboard colourable ribbon graph minors

Motivated by the Eulerian ribbon graph minors, in this paper we introduce the notion of checkerboard colourable minors for ribbon graphs and its dual: bipartite minors for ribbon graphs. Motivated by the bipartite minors of abstract graphs, another bipartite minors for ribbon graphs, i.e. the bipartite ribbon graph join minors are also introduced. Using these minors then we give excluded minor characterizations of the classes of checkerboard colourable ribbon graphs, bipartite ribbon graphs, plane checkerboard colourable ribbon graphs and plane bipartite ribbon graphs.     

A characterization of triangle-free tolerance graphs

We prove that a triangle-free graph G is a tolerance graph if and only if there exists a set of consecutively ordered stars that partition the edges of G. Since tolerance graphs are weakly chordal, a tolerance graph is bipartite if and only if it is triangle-free. We, therefore, characterize those tolerance graphs that are also bipartite. We use this result to show that in general, the class of interval bigraphs properly contains tolerance graphs that are triangle-free (and hence bipartite).     

Spanning trees with pairwise nonadjacent endvertices

A spanning tree of a connected graph G is said to be an independency tree if all its endvertices are pairwise nonadjacent in G. We prove that a connected graph G has no independency tree if and only if G is a cycle, a complete graph or a complete bipartite graph the color classes of which have equal cardinality.     

Hamiltonicity in vertex envelopes of plane cubic graphs

In this paper we study a graph operation which produces what we call the “vertex envelope” G^∗_V from a graph G. We apply it to plane cubic graphs and investigate the hamiltonicity of the resulting graphs, which are also cubic. To this end, we prove a result giving a necessary and sufficient condition for the existence of hamiltonian cycles in the vertex envelopes of plane cubic graphs. We then use these conditions to identify graphs or classes of graphs whose vertex envelopes are either all hamiltonian or all non-hamiltonian, paying special attention to bipartite graphs. We also show that deciding if a vertex envelope is hamiltonian is NP-complete, and we provide a polynomial algorithm for deciding if a given cubic plane graph is a vertex envelope.     

On encodings of spanning trees

Deo and Micikevicius recently gave a new bijection for spanning trees of complete bipartite graphs. In this paper we devise a generalization of Deo and Micikevicius's method, which is also a modification of Olah's method for encoding the spanning trees of any complete multipartite graph K(n₁,…,n_r). We also give a bijection between the spanning trees of a planar graph and those of any of its planar duals. Finally we discuss the possibility of bijections for spanning trees of DeBriujn graphs, cubes, and regular graphs such as the Petersen graph that have integer eigenvalues.     

Crossing graphs of fiber-complemented graphs

Fiber-complemented graphs form a vast non bipartite generalization of median graphs. Using a certain natural coloring of edges, induced by parallelism relation between prefibers of a fiber-complemented graph, we introduce the crossing graph of a fiber-complemented graph G as the graph whose vertices are colors, and two colors are adjacent if they cross on some induced 4-cycle in G. We show that a fiber-complemented graph is 2-connected if and only if its crossing graph is connected. We characterize those fiber-complemented graphs whose crossing graph is complete, and also those whose crossing graph is chordal.     

Cell rotation graphs of strongly connected orientations of plane graphs with an application

The cell rotation graph D(G) on the strongly connected orientations of a 2-edge-connected plane graph G is defined. It is shown that D(G) is a directed forest and every component is an in-tree with one root; if T is a component of D(G), the reversions of all orientations in T induce a component of D(G), denoted by T⁻, thus (T,T⁻) is called a pair of in-trees of D(G); G is Eulerian if and only if D(G) has an odd number of components (all Eulerian orientations of G induce the same component of D(G)); the width and height of T are equal to that of T⁻, respectively. Further it is shown that the pair of directed tree structures on the perfect matchings of a plane elementary bipartite graph G coincide with a pair of in-trees of D(G). Accordingly, such a pair of in-trees on the perfect matchings of any plane bipartite graph have the same width and height.     

Turán-type results for partial orders and intersection graphs of convex sets

We prove Ramsey-type results for intersection graphs of geometric objects. In particular, we prove the following bounds, all of which are tight apart from the constant c. There is a constant c > 0 such that for every family F of n convex sets in the plane, the intersection graph of F or its complement contains a balanced complete bipartite graph of size at least cn. There is a constant c > 0 such that for every family F of n x-monotone curves in the plane, the intersection graph G of F contains a balanced complete bipartite graph of size at least cn/log n or the complement of G contains a balanced complete bipartite graph of size at least cn. Our bounds rely on new Turán-type results on incomparability graphs of partially ordered sets.     

The uniquely embeddable planar graphs

Whitney [7] proved in 1932 that for any two embeddings of a planar 3-connected graph, their combinatorial duals are isomorphic. In this manner, the term “uniquely embeddable planar graph” was introduced. It is a well-known fact that combinatorial and geometrical duals are equivalent concepts. In this paper, the concept of unique embeddability is introduced in terms of special types of isomorphisms between any two embeddings of a planar graph. From this, the class U of all graphs which are uniquely embeddable in the plane according to this definition, is determined, and the planar 3-connected graphs are a proper subset of U. It turns out that the graphs in U have a unique geometrical dual (i.e., for any two embeddings of such a graph, their geometrical duals are isomorphic). Furthermore, the theorems and their proofs do not involve any type of duals.     

On hereditary clique-Helly self-clique graphs

A graph is clique-Helly if any family of mutually intersecting (maximal) cliques has non-empty intersection, and it is hereditary clique-Helly (HCH) if its induced subgraphs are clique-Helly. The clique graph of a graph G is the intersection graph of its cliques, and G is self-clique if it is connected and isomorphic to its clique graph. We show that every HCH graph is an induced subgraph of a self-clique HCH graph, and give a characterization of self-clique HCH graphs in terms of their constructibility starting from certain digraphs with some forbidden subdigraphs. We also specialize this results to involutive HCH graphs, i.e. self-clique HCH graphs whose vertex-clique bipartite graph admits a part-switching involution.     

A note on Barnette’s conjecture

A well-known conjecture of Barnette states that every 3-connected cubic bipartite planar graph has a Hamiltonian cycle, which is equivalent to the statement that every 3-connected even plane triangulation admits a 2-tree coloring, meaning that the vertices of the graph have a 2-coloring such that each color class induces a tree. In this paper we present a new approach to Barnette’s conjecture by using 2-tree coloring.A Barnette triangulation is a 3-connected even plane triangulation, and a B-graph is a smallest Barnette triangulation without a 2-tree coloring. A configuration is reducible if it cannot be a configuration of a B-graph. We prove that certain configurations are reducible. We also define extendable, non-extendable and compatible graphs; and discuss their connection with Barnette’s conjecture.     

Maximum order of trees and bipartite graphs with a given rank

The rank of a graph is that of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. We determine the maximum order of reduced trees as well as bipartite graphs with a given rank and characterize those graphs achieving the maximum order.